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KingGeorge

  • 3 years ago

[SOLVED] Here's a cute predicate logic problem. Prove or disprove: \[\forall x \,\exists y \,\forall z\, \exists w \;\;\text{such that} \;\;x+y+z+w=xyzw\]Assume that \(x, y, z, w \in \mathbb{C}\).

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  1. experimentX
    • 3 years ago
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    is xyzw a combination of digit or multiplication of digit??

  2. KingGeorge
    • 3 years ago
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    multiplication

  3. experimentX
    • 3 years ago
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    lol .. didn't realize they were complex

  4. KingGeorge
    • 3 years ago
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    It doesn't really matter. I just put that specification on there so that know one tried to change the problem so that addition and multiplication had different meanings.

  5. KingGeorge
    • 3 years ago
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    Just prove/disprove it for \(\mathbb{R}\) and it will become clear why it doesn't matter if they're complex numbers.

  6. KingGeorge
    • 3 years ago
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    Hint below in white LaTeX\[\color{white}{\text{There is a certain value for y that you should choose to prove it.}}\]

  7. experimentX
    • 3 years ago
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    \[ \color{white}{\text{LOL i didn't know that!!.}} \]

  8. KingGeorge
    • 3 years ago
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    Another, probably more helpful hint, below in white LaTeX\[\color{white}{\text{What value of y can you choose so that xyzw is constant?}}\]

  9. nbouscal
    • 3 years ago
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    Choose y=0, then choose w=-(x+z).

  10. KingGeorge
    • 3 years ago
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    Bingo.

  11. experimentX
    • 3 years ago
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    But ∃y doesn't mean that y is rather a variable such that we have to fit constraint ??

  12. KingGeorge
    • 3 years ago
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    \(\exists\, y\) means that we can choose a specific value for y. It's a variable, but one that we have complete control over.

  13. experimentX
    • 3 years ago
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    then it must be false ... i guess so!!

  14. KingGeorge
    • 3 years ago
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    The "there exists" only means that at least one possible value of y where this is true. One value that we know works is \(y=0\). This means that there does exists at least one value for \(y\) that makes the statement true, so we have proven that the original statement is true.

  15. experimentX
    • 3 years ago
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    does't y=0 make statement untrue?? xyzw=0, for y=0

  16. nbouscal
    • 3 years ago
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    Right, both the sum and the multiplication end up equaling zero because you set w=-(x+z).

  17. KingGeorge
    • 3 years ago
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    If \(y\) is 0, and \(x=-(x+z)\) we have \[x+0+z+(-(x+z))=x+z-x-z=0\]and \[x\cdot0\cdot z\cdot w=0\]So that means that \[x+y+z+w=xyzw\]

  18. experimentX
    • 3 years ago
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    Oh ... stupid me ... i never considered the negative values!!

  19. KingGeorge
    • 3 years ago
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    Hope everyone enjoyed this little (unintuitive) problem. Good night.

  20. experimentX
    • 3 years ago
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    thanks!! i'll try to think broader way next time!!

  21. KingGeorge
    • 3 years ago
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    I'll try to get a problem just for you next time then. Any preference about the subject?

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